Properties

Label 8034.2.a.v
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{9} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{9} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + \beta_{1} q^{10} \) \( + ( -1 + \beta_{4} ) q^{11} \) \(+ q^{12}\) \(+ q^{13}\) \( -\beta_{9} q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( 1 - \beta_{2} ) q^{17} \) \(+ q^{18}\) \( + ( -1 - \beta_{9} ) q^{19} \) \( + \beta_{1} q^{20} \) \( -\beta_{9} q^{21} \) \( + ( -1 + \beta_{4} ) q^{22} \) \( + ( 2 + \beta_{5} ) q^{23} \) \(+ q^{24}\) \( + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{25} \) \(+ q^{26}\) \(+ q^{27}\) \( -\beta_{9} q^{28} \) \( + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} \) \( + \beta_{1} q^{30} \) \( + ( 1 + \beta_{9} ) q^{31} \) \(+ q^{32}\) \( + ( -1 + \beta_{4} ) q^{33} \) \( + ( 1 - \beta_{2} ) q^{34} \) \( + ( 2 + \beta_{3} - \beta_{5} - \beta_{9} ) q^{35} \) \(+ q^{36}\) \( + ( 2 - \beta_{3} + \beta_{4} + \beta_{9} ) q^{37} \) \( + ( -1 - \beta_{9} ) q^{38} \) \(+ q^{39}\) \( + \beta_{1} q^{40} \) \( + ( -1 - \beta_{5} - \beta_{7} ) q^{41} \) \( -\beta_{9} q^{42} \) \( + ( -1 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{43} \) \( + ( -1 + \beta_{4} ) q^{44} \) \( + \beta_{1} q^{45} \) \( + ( 2 + \beta_{5} ) q^{46} \) \( + ( 1 + \beta_{3} ) q^{47} \) \(+ q^{48}\) \( + ( 3 - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{49} \) \( + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{50} \) \( + ( 1 - \beta_{2} ) q^{51} \) \(+ q^{52}\) \( + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{53} \) \(+ q^{54}\) \( + ( \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{55} \) \( -\beta_{9} q^{56} \) \( + ( -1 - \beta_{9} ) q^{57} \) \( + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{58} \) \( + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{61} \) \( + ( 1 + \beta_{9} ) q^{62} \) \( -\beta_{9} q^{63} \) \(+ q^{64}\) \( + \beta_{1} q^{65} \) \( + ( -1 + \beta_{4} ) q^{66} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{67} \) \( + ( 1 - \beta_{2} ) q^{68} \) \( + ( 2 + \beta_{5} ) q^{69} \) \( + ( 2 + \beta_{3} - \beta_{5} - \beta_{9} ) q^{70} \) \( + ( 3 + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{71} \) \(+ q^{72}\) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{73} \) \( + ( 2 - \beta_{3} + \beta_{4} + \beta_{9} ) q^{74} \) \( + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{75} \) \( + ( -1 - \beta_{9} ) q^{76} \) \( + ( 4 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{77} \) \(+ q^{78}\) \( + ( 5 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} \) \( + \beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -1 - \beta_{5} - \beta_{7} ) q^{82} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{83} \) \( -\beta_{9} q^{84} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} ) q^{85} \) \( + ( -1 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{86} \) \( + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{87} \) \( + ( -1 + \beta_{4} ) q^{88} \) \( + ( 2 - \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{89} \) \( + \beta_{1} q^{90} \) \( -\beta_{9} q^{91} \) \( + ( 2 + \beta_{5} ) q^{92} \) \( + ( 1 + \beta_{9} ) q^{93} \) \( + ( 1 + \beta_{3} ) q^{94} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{9} ) q^{95} \) \(+ q^{96}\) \( + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{97} \) \( + ( 3 - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{98} \) \( + ( -1 + \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 11q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 11q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 11q^{8} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 11q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 11q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 11q^{8} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 11q^{12} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 11q^{16} \) \(\mathstrut +\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 11q^{18} \) \(\mathstrut -\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 5q^{20} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 7q^{22} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 11q^{24} \) \(\mathstrut +\mathstrut 32q^{25} \) \(\mathstrut +\mathstrut 11q^{26} \) \(\mathstrut +\mathstrut 11q^{27} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 29q^{29} \) \(\mathstrut +\mathstrut 5q^{30} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 11q^{32} \) \(\mathstrut -\mathstrut 7q^{33} \) \(\mathstrut +\mathstrut 10q^{34} \) \(\mathstrut +\mathstrut 31q^{35} \) \(\mathstrut +\mathstrut 11q^{36} \) \(\mathstrut +\mathstrut 21q^{37} \) \(\mathstrut -\mathstrut 7q^{38} \) \(\mathstrut +\mathstrut 11q^{39} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 17q^{43} \) \(\mathstrut -\mathstrut 7q^{44} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut +\mathstrut 18q^{46} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut +\mathstrut 11q^{48} \) \(\mathstrut +\mathstrut 21q^{49} \) \(\mathstrut +\mathstrut 32q^{50} \) \(\mathstrut +\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 11q^{52} \) \(\mathstrut +\mathstrut 11q^{53} \) \(\mathstrut +\mathstrut 11q^{54} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 7q^{57} \) \(\mathstrut +\mathstrut 29q^{58} \) \(\mathstrut -\mathstrut 48q^{59} \) \(\mathstrut +\mathstrut 5q^{60} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut 7q^{62} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 11q^{64} \) \(\mathstrut +\mathstrut 5q^{65} \) \(\mathstrut -\mathstrut 7q^{66} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut +\mathstrut 10q^{68} \) \(\mathstrut +\mathstrut 18q^{69} \) \(\mathstrut +\mathstrut 31q^{70} \) \(\mathstrut +\mathstrut 17q^{71} \) \(\mathstrut +\mathstrut 11q^{72} \) \(\mathstrut -\mathstrut 23q^{73} \) \(\mathstrut +\mathstrut 21q^{74} \) \(\mathstrut +\mathstrut 32q^{75} \) \(\mathstrut -\mathstrut 7q^{76} \) \(\mathstrut +\mathstrut 26q^{77} \) \(\mathstrut +\mathstrut 11q^{78} \) \(\mathstrut +\mathstrut 41q^{79} \) \(\mathstrut +\mathstrut 5q^{80} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut -\mathstrut 3q^{82} \) \(\mathstrut +\mathstrut 19q^{83} \) \(\mathstrut +\mathstrut 4q^{84} \) \(\mathstrut +\mathstrut 17q^{85} \) \(\mathstrut -\mathstrut 17q^{86} \) \(\mathstrut +\mathstrut 29q^{87} \) \(\mathstrut -\mathstrut 7q^{88} \) \(\mathstrut +\mathstrut 32q^{89} \) \(\mathstrut +\mathstrut 5q^{90} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 18q^{92} \) \(\mathstrut +\mathstrut 7q^{93} \) \(\mathstrut +\mathstrut 12q^{94} \) \(\mathstrut +\mathstrut 26q^{95} \) \(\mathstrut +\mathstrut 11q^{96} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut 21q^{98} \) \(\mathstrut -\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(5\) \(x^{10}\mathstrut -\mathstrut \) \(31\) \(x^{9}\mathstrut +\mathstrut \) \(168\) \(x^{8}\mathstrut +\mathstrut \) \(300\) \(x^{7}\mathstrut -\mathstrut \) \(1928\) \(x^{6}\mathstrut -\mathstrut \) \(736\) \(x^{5}\mathstrut +\mathstrut \) \(8532\) \(x^{4}\mathstrut -\mathstrut \) \(2065\) \(x^{3}\mathstrut -\mathstrut \) \(10494\) \(x^{2}\mathstrut +\mathstrut \) \(4024\) \(x\mathstrut +\mathstrut \) \(576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(7214441\) \(\nu^{10}\mathstrut -\mathstrut \) \(17965281\) \(\nu^{9}\mathstrut +\mathstrut \) \(354406289\) \(\nu^{8}\mathstrut +\mathstrut \) \(1471530490\) \(\nu^{7}\mathstrut -\mathstrut \) \(7557802336\) \(\nu^{6}\mathstrut -\mathstrut \) \(29633121208\) \(\nu^{5}\mathstrut +\mathstrut \) \(75666608448\) \(\nu^{4}\mathstrut +\mathstrut \) \(192625837968\) \(\nu^{3}\mathstrut -\mathstrut \) \(292454456923\) \(\nu^{2}\mathstrut -\mathstrut \) \(278709400064\) \(\nu\mathstrut +\mathstrut \) \(159253845372\)\()/\)\(24169157748\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(6954475\) \(\nu^{10}\mathstrut +\mathstrut \) \(45806123\) \(\nu^{9}\mathstrut +\mathstrut \) \(154221401\) \(\nu^{8}\mathstrut -\mathstrut \) \(1300710052\) \(\nu^{7}\mathstrut -\mathstrut \) \(1247220084\) \(\nu^{6}\mathstrut +\mathstrut \) \(11904407208\) \(\nu^{5}\mathstrut +\mathstrut \) \(10889091824\) \(\nu^{4}\mathstrut -\mathstrut \) \(42269912108\) \(\nu^{3}\mathstrut -\mathstrut \) \(58601234445\) \(\nu^{2}\mathstrut +\mathstrut \) \(68038889518\) \(\nu\mathstrut +\mathstrut \) \(3291397224\)\()/\)\(16112771832\)
\(\beta_{4}\)\(=\)\((\)\(12175783\) \(\nu^{10}\mathstrut -\mathstrut \) \(58843755\) \(\nu^{9}\mathstrut -\mathstrut \) \(96960025\) \(\nu^{8}\mathstrut +\mathstrut \) \(1153915696\) \(\nu^{7}\mathstrut -\mathstrut \) \(4060162144\) \(\nu^{6}\mathstrut -\mathstrut \) \(2903894740\) \(\nu^{5}\mathstrut +\mathstrut \) \(53455821804\) \(\nu^{4}\mathstrut -\mathstrut \) \(26795724084\) \(\nu^{3}\mathstrut -\mathstrut \) \(160415090299\) \(\nu^{2}\mathstrut +\mathstrut \) \(34242147922\) \(\nu\mathstrut +\mathstrut \) \(68016964824\)\()/\)\(24169157748\)
\(\beta_{5}\)\(=\)\((\)\(6501563\) \(\nu^{10}\mathstrut +\mathstrut \) \(42249738\) \(\nu^{9}\mathstrut -\mathstrut \) \(439905770\) \(\nu^{8}\mathstrut -\mathstrut \) \(1187809591\) \(\nu^{7}\mathstrut +\mathstrut \) \(8087991802\) \(\nu^{6}\mathstrut +\mathstrut \) \(11048729236\) \(\nu^{5}\mathstrut -\mathstrut \) \(51187504290\) \(\nu^{4}\mathstrut -\mathstrut \) \(39173547342\) \(\nu^{3}\mathstrut +\mathstrut \) \(83266816921\) \(\nu^{2}\mathstrut +\mathstrut \) \(42230737499\) \(\nu\mathstrut +\mathstrut \) \(16835685894\)\()/\)\(12084578874\)
\(\beta_{6}\)\(=\)\((\)\(6501563\) \(\nu^{10}\mathstrut +\mathstrut \) \(42249738\) \(\nu^{9}\mathstrut -\mathstrut \) \(439905770\) \(\nu^{8}\mathstrut -\mathstrut \) \(1187809591\) \(\nu^{7}\mathstrut +\mathstrut \) \(8087991802\) \(\nu^{6}\mathstrut +\mathstrut \) \(11048729236\) \(\nu^{5}\mathstrut -\mathstrut \) \(51187504290\) \(\nu^{4}\mathstrut -\mathstrut \) \(39173547342\) \(\nu^{3}\mathstrut +\mathstrut \) \(95351395795\) \(\nu^{2}\mathstrut +\mathstrut \) \(30146158625\) \(\nu\mathstrut -\mathstrut \) \(67756366224\)\()/\)\(12084578874\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(64472003\) \(\nu^{10}\mathstrut +\mathstrut \) \(67818039\) \(\nu^{9}\mathstrut +\mathstrut \) \(2616704621\) \(\nu^{8}\mathstrut -\mathstrut \) \(1691813840\) \(\nu^{7}\mathstrut -\mathstrut \) \(36952460260\) \(\nu^{6}\mathstrut +\mathstrut \) \(10235837864\) \(\nu^{5}\mathstrut +\mathstrut \) \(196673190432\) \(\nu^{4}\mathstrut +\mathstrut \) \(3251983908\) \(\nu^{3}\mathstrut -\mathstrut \) \(197764359613\) \(\nu^{2}\mathstrut -\mathstrut \) \(33353267558\) \(\nu\mathstrut -\mathstrut \) \(332744495184\)\()/\)\(48338315496\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(13289341\) \(\nu^{10}\mathstrut +\mathstrut \) \(82204613\) \(\nu^{9}\mathstrut +\mathstrut \) \(372751607\) \(\nu^{8}\mathstrut -\mathstrut \) \(2717567812\) \(\nu^{7}\mathstrut -\mathstrut \) \(2917748212\) \(\nu^{6}\mathstrut +\mathstrut \) \(30309554056\) \(\nu^{5}\mathstrut +\mathstrut \) \(1318194840\) \(\nu^{4}\mathstrut -\mathstrut \) \(127738310148\) \(\nu^{3}\mathstrut +\mathstrut \) \(40432304461\) \(\nu^{2}\mathstrut +\mathstrut \) \(142993848946\) \(\nu\mathstrut -\mathstrut \) \(20159257776\)\()/\)\(5370923944\)
\(\beta_{9}\)\(=\)\((\)\(125904587\) \(\nu^{10}\mathstrut -\mathstrut \) \(456748671\) \(\nu^{9}\mathstrut -\mathstrut \) \(4328210285\) \(\nu^{8}\mathstrut +\mathstrut \) \(14601473048\) \(\nu^{7}\mathstrut +\mathstrut \) \(51523740940\) \(\nu^{6}\mathstrut -\mathstrut \) \(155126675336\) \(\nu^{5}\mathstrut -\mathstrut \) \(239310756048\) \(\nu^{4}\mathstrut +\mathstrut \) \(597489887604\) \(\nu^{3}\mathstrut +\mathstrut \) \(307612462405\) \(\nu^{2}\mathstrut -\mathstrut \) \(504759302554\) \(\nu\mathstrut -\mathstrut \) \(33312963024\)\()/\)\(48338315496\)
\(\beta_{10}\)\(=\)\((\)\(11080037\) \(\nu^{10}\mathstrut -\mathstrut \) \(37445815\) \(\nu^{9}\mathstrut -\mathstrut \) \(379154155\) \(\nu^{8}\mathstrut +\mathstrut \) \(1228551980\) \(\nu^{7}\mathstrut +\mathstrut \) \(4492255116\) \(\nu^{6}\mathstrut -\mathstrut \) \(14049457206\) \(\nu^{5}\mathstrut -\mathstrut \) \(21039436504\) \(\nu^{4}\mathstrut +\mathstrut \) \(64779403690\) \(\nu^{3}\mathstrut +\mathstrut \) \(28287160275\) \(\nu^{2}\mathstrut -\mathstrut \) \(88710185906\) \(\nu\mathstrut +\mathstrut \) \(9170866704\)\()/\)\(4028192958\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(79\)
\(\nu^{5}\)\(=\)\(-\)\(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(28\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(44\) \(\beta_{6}\mathstrut -\mathstrut \) \(22\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(49\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(168\) \(\beta_{1}\mathstrut +\mathstrut \) \(47\)
\(\nu^{6}\)\(=\)\(-\)\(29\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{8}\mathstrut -\mathstrut \) \(117\) \(\beta_{7}\mathstrut +\mathstrut \) \(282\) \(\beta_{6}\mathstrut -\mathstrut \) \(326\) \(\beta_{5}\mathstrut -\mathstrut \) \(27\) \(\beta_{4}\mathstrut -\mathstrut \) \(79\) \(\beta_{3}\mathstrut +\mathstrut \) \(56\) \(\beta_{2}\mathstrut +\mathstrut \) \(391\) \(\beta_{1}\mathstrut +\mathstrut \) \(1035\)
\(\nu^{7}\)\(=\)\(-\)\(274\) \(\beta_{10}\mathstrut -\mathstrut \) \(341\) \(\beta_{9}\mathstrut -\mathstrut \) \(240\) \(\beta_{8}\mathstrut -\mathstrut \) \(350\) \(\beta_{7}\mathstrut +\mathstrut \) \(842\) \(\beta_{6}\mathstrut -\mathstrut \) \(448\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut -\mathstrut \) \(979\) \(\beta_{3}\mathstrut +\mathstrut \) \(209\) \(\beta_{2}\mathstrut +\mathstrut \) \(2537\) \(\beta_{1}\mathstrut +\mathstrut \) \(1196\)
\(\nu^{8}\)\(=\)\(-\)\(677\) \(\beta_{10}\mathstrut -\mathstrut \) \(121\) \(\beta_{9}\mathstrut +\mathstrut \) \(244\) \(\beta_{8}\mathstrut -\mathstrut \) \(2165\) \(\beta_{7}\mathstrut +\mathstrut \) \(4726\) \(\beta_{6}\mathstrut -\mathstrut \) \(5433\) \(\beta_{5}\mathstrut -\mathstrut \) \(451\) \(\beta_{4}\mathstrut -\mathstrut \) \(2044\) \(\beta_{3}\mathstrut +\mathstrut \) \(1277\) \(\beta_{2}\mathstrut +\mathstrut \) \(7030\) \(\beta_{1}\mathstrut +\mathstrut \) \(14810\)
\(\nu^{9}\)\(=\)\(-\)\(5611\) \(\beta_{10}\mathstrut -\mathstrut \) \(3868\) \(\beta_{9}\mathstrut -\mathstrut \) \(2996\) \(\beta_{8}\mathstrut -\mathstrut \) \(6017\) \(\beta_{7}\mathstrut +\mathstrut \) \(15514\) \(\beta_{6}\mathstrut -\mathstrut \) \(8891\) \(\beta_{5}\mathstrut +\mathstrut \) \(395\) \(\beta_{4}\mathstrut -\mathstrut \) \(18203\) \(\beta_{3}\mathstrut +\mathstrut \) \(5188\) \(\beta_{2}\mathstrut +\mathstrut \) \(40156\) \(\beta_{1}\mathstrut +\mathstrut \) \(25238\)
\(\nu^{10}\)\(=\)\(-\)\(14206\) \(\beta_{10}\mathstrut +\mathstrut \) \(969\) \(\beta_{9}\mathstrut +\mathstrut \) \(6288\) \(\beta_{8}\mathstrut -\mathstrut \) \(37184\) \(\beta_{7}\mathstrut +\mathstrut \) \(80285\) \(\beta_{6}\mathstrut -\mathstrut \) \(89691\) \(\beta_{5}\mathstrut -\mathstrut \) \(5778\) \(\beta_{4}\mathstrut -\mathstrut \) \(45119\) \(\beta_{3}\mathstrut +\mathstrut \) \(26759\) \(\beta_{2}\mathstrut +\mathstrut \) \(124641\) \(\beta_{1}\mathstrut +\mathstrut \) \(224907\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.73039
−3.11723
−2.84496
−1.31499
−0.111668
0.542814
1.54900
2.64390
3.14620
4.01421
4.22313
1.00000 1.00000 1.00000 −3.73039 1.00000 −3.81724 1.00000 1.00000 −3.73039
1.2 1.00000 1.00000 1.00000 −3.11723 1.00000 −0.395748 1.00000 1.00000 −3.11723
1.3 1.00000 1.00000 1.00000 −2.84496 1.00000 −0.764995 1.00000 1.00000 −2.84496
1.4 1.00000 1.00000 1.00000 −1.31499 1.00000 3.43679 1.00000 1.00000 −1.31499
1.5 1.00000 1.00000 1.00000 −0.111668 1.00000 −0.538332 1.00000 1.00000 −0.111668
1.6 1.00000 1.00000 1.00000 0.542814 1.00000 3.05565 1.00000 1.00000 0.542814
1.7 1.00000 1.00000 1.00000 1.54900 1.00000 −5.16602 1.00000 1.00000 1.54900
1.8 1.00000 1.00000 1.00000 2.64390 1.00000 4.97730 1.00000 1.00000 2.64390
1.9 1.00000 1.00000 1.00000 3.14620 1.00000 1.95604 1.00000 1.00000 3.14620
1.10 1.00000 1.00000 1.00000 4.01421 1.00000 2.23635 1.00000 1.00000 4.01421
1.11 1.00000 1.00000 1.00000 4.22313 1.00000 −0.979791 1.00000 1.00000 4.22313
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)